Finite Group Databasebeta

← #182649

Subgroups — #182649

Order 512.

1556 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34204C211yesmaximal, characteristic
2#20379C211yesmaximal, characteristic, complemented
3C2×#1519C211yesmaximal, characteristic
4C2×#1507C211yesmaximal, characteristic
5#13801C211yesmaximal, characteristic, complemented
6#8220C212yesmaximal, complemented
7#8218C212yesmaximal, complemented
8#8216C212yesmaximal, complemented
9C4×#327C212yesmaximal, complemented
10#8016C211yesmaximal, characteristic, complemented
11#8012C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C2×#518C411yescharacteristic, complemented
14C22×#117C2211yescharacteristic
15C2×#407C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C2×#405—21no—
18#2363—21no—
19#2363—21no—
20#2361—21no—
21#2361—21no—
22#2342—22no—
23D4×C22⋊C4—22no—
24#2324C412yescomplemented
25#2324C412yescomplemented
26#2316C411yescharacteristic, complemented
27#2309C411yescharacteristic, complemented
28C4×#121C2212yes—
29C4×#121C2212yes—
30#2280C2211yescharacteristic
31C2×#380C2211yescharacteristic
32#2274C411yescharacteristic, complemented
33C2×C4×C4⋊C4—21no—
34C2×C4×C22⋊C4C2211yescharacteristic
35#1750—22no—
36#1749C2212yescomplemented
37C4×C22⋊C8C2212yescomplemented
38#1729—21no—
39#1519C418yescomplemented
40#1507C418yescomplemented
41C2×#327C2212yes—
42C2×#327C2212yes—
43C2×#327C2212yes—
44C2×#327C2212yes—
45#1437—22no—
46C4⋊#95—21no—
47#1434C2212yescomplemented
48C4×#95C2212yescomplemented
49#1294C2211yescharacteristic, complemented
50#1294C2212yescomplemented
51#1294C2211yescharacteristic, complemented
52#1294C2211yescharacteristic, complemented
53#1294C2212yescomplemented
54#1294C2211yescharacteristic, complemented
55C23×D4—21no—
56C23×D4C2×C411yescharacteristic
57C24×C4C2311yescharacteristic
58C22×M16—21no—
59C23×C8—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121C2×C414yes—
65C2×#121C2×C414yes—
66C2×#121C2×C412yes—
67C2×#121C2312yes—
68#518—44no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2×C412yes—
74C2×C4×D4C2×C412yes—
75C2×#117—44no—
76C2×#117C2×C411yescharacteristic
77C2×#117C2×C411yescharacteristic
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C412yes—
81C2×#117D411yescharacteristic
82C2×#117D412yes—
83C2×#117D411yescharacteristic
84C22×C4⋊C4—21no—
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C4⋊C4D411yescharacteristic
87C22×C4⋊C4D411yescharacteristic
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4D412yes—
93C22×C42—21no—
94C22×C42—22no—
95C22×C42C2312yes—
96#407—22no—
97#407C2×C414yescomplemented
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8C2×C414yescomplemented
102C2×C22⋊C8C2311yescharacteristic
103C2×C22⋊C8C2311yescharacteristic
104C2×C22⋊C8C2312yes—
105#405—44no—
106C2×#97—22no—
107C2×C4×C8—22no—
108#390—21no—
109#390—21no—
110#390—22no—
111#390—22no—
112#390C2×C412yes—
113#390C2×C412yes—
114#389—21no—
115#389—21no—
116#389C2×C412yes—
117#389C2×C412yes—
118#388—22no—
119#388—22no—
120#388—22no—
121#388—22no—
122#386—22no—
123#386—22no—
124#386—22no—
125#386—22no—
126#385—22no—
127#385—22no—
128#381—21no—
129#380—22no—
130#380—22no—
131#380C2×C411yescharacteristic
132#380C2×C411yescharacteristic
133#380C2×C414yescomplemented
134#380C2311yescharacteristic
135#380C2311yescharacteristic
136#379—21no—
137#379—22no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—44no—
141C4×C4⋊C4C2×C412yes—
142C4×C4⋊C4C2×C412yes—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149C4×C22⋊C4C2×C414yescomplemented
150C2×#95—22no—
151C2×#95—22no—
152C2×#95—22no—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156#336—22no—
157#336—22no—
158#327C2×C418yescomplemented
159#327C2×C418yescomplemented
160#327C2×C418yescomplemented
161#327C2×C418yescomplemented
162C25—21no—
163C25C2×D411yescharacteristic
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4—81no—
173C22×D4—81no—
174C22×D4—81no—
175C22×D4—81no—
176C22×D4C4214yescomplemented
177C22×D4C22×C412yes—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—22no—
181C23×C4—21no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C412yes—
189C23×C4C22×C412yes—
190C23×C4C2×D412yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×D411yescharacteristic
195C23×C4#4112yes—
196C23×C4#4112yes—
197C23×C4C2411yescharacteristic
198C2×M16—24no—
199C2×M16—42no—
200C2×M16—42no—
201C22×C8—24no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8—42no—
206C22×C8—42no—
207#121—24no—
208#121—24no—
209#121—24no—
210#121—24no—
211#121—24no—
212#121—22no—
213#121—22no—
214#121C4218yescomplemented
215#121C22×C414yes—
216#121C22×C414yes—
217C4×D4—44no—
218C4×D4—44no—
219C4×D4—44no—
220C4×D4—44no—
221C4×D4—82no—
222C4×D4—82no—
223C4×D4—82no—
224C4×D4—82no—
225#117—24no—
226#117—24no—
227#117—44no—
228#117—44no—
229#117—42no—
230#117—42no—
231#117—42no—
232#117—42no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4C4214yescomplemented
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22⋊C412yes—
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22⋊C412yes—
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—81no—
277C2×C22⋊C4—81no—
278C2×C22⋊C4C22⋊C412yes—
279C2×C22⋊C4C22⋊C412yes—
280C2×C22⋊C4C22⋊C412yes—
281C2×C22⋊C4C22⋊C412yes—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—44no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—82no—
293C2×C42C22×C412yes—
294C2×C42C22×C412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C22⋊C8—24no—
298C22⋊C8—24no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—22no—
302C22⋊C8C22×C412yes—
303C22⋊C8C22×C412yes—
304C22⋊C8C22×C414yes—
305#97—44no—
306C4×C8—44no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—22no—
318#95—44no—
319#95—44no—
320#95—82no—
321#95C22×C412yes—
322#95C22×C412yes—
323#95C22×C414yes—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—22no—
329C24—24no—
330C24—41no—
331C24—41no—
332C24—41no—
333C24—41no—
334C24—42no—
335C24—42no—
336C24—81no—
337C24—81no—
338C24—81no—
339C24—81no—
340C24C2×C22⋊C411yescharacteristic
341C24C2×C22⋊C411yescharacteristic
342C24C4×D412yes—
343C24C4×D412yes—
344C24C22×D411yescharacteristic
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—28no—
348C2×D4—22no—
349C2×D4—24no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—82no—
361C2×D4—82no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4C2×C4214yes—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—41no—
419C22×C4—44no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4C2×C4212yes—
427C22×C4C2×C4212yes—
428C22×C4C2×C22⋊C412yes—
429C22×C4C2×C22⋊C412yes—
430C22×C4C2×C22⋊C412yes—
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C412yes—
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4#11712yes—
435C22×C4#11712yes—
436C22×C4C4×D412yes—
437C22×C4C4×D412yes—
438C22×C4C4×D412yes—
439C22×C4C4×D412yes—
440C22×C4C4×D412yes—
441C22×C4C4×D412yes—
442C22×C4C23×C411yescharacteristic
443C22×C4C23×C411yescharacteristic
444C22×C4C23×C411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448M16—48no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—41no—
453C2×C8—42no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—48no—
458C2×C8—42no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—48no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4#9914yes—
474C4⋊C4C2×C4214yes—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—28no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—48no—
483C22⋊C4—44no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—44no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—81no—
496C22⋊C4—81no—
497C22⋊C4—84no—
498C42—24no—
499C42—24no—
500C42—42no—
501C42—42no—
502C42—48no—
503C42—42no—
504C42—42no—
505C42—82no—
506C42—82no—
507C42—88no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—48no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—81no—
541C23—81no—
542C23—81no—
543C23—81no—
544C23—81no—
545C23—81no—
546C23—81no—
547C23—81no—
548C23—84no—
549C23—84no—
550C23—162no—
551C23—162no—
552C23—162no—
553C23—162no—
554C23C4×C22⋊C412yes—
555C23C4×C22⋊C412yes—
556C23C22×C22⋊C411yescharacteristic
557C23C2×C4×D411yescharacteristic
558C23C2×C4×D411yescharacteristic
559C23#57611yescharacteristic
560C23#57611yescharacteristic
561C23#57711yescharacteristic
562C23#57711yescharacteristic
563D4—44no—
564D4—44no—
565D4—44no—
566D4—44no—
567D4—44no—
568D4—44no—
569D4—48no—
570D4—88no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—84no—
635C2×C4—82no—
636C2×C4—81no—
637C2×C4—82no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—82no—
641C2×C4—81no—
642C2×C4—82no—
643C2×C4—164no—
644C2×C4—164no—
645C2×C4C4×C22⋊C412yes—
646C2×C4C4×C22⋊C412yes—
647C2×C4C2×#9914yes—
648C2×C4C2×#9912yes—
649C2×C4#41012yes—
650C2×C4C22×C4211yescharacteristic
651C2×C4C2×#11711yescharacteristic
652C2×C4C2×C4×D411yescharacteristic
653C2×C4C2×C4×D411yescharacteristic
654C8—42no—
655C8—42no—
656C8—44no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—21no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—168no—
695C22C2×C4×C22⋊C411yescharacteristic
696C22C22×#9911yescharacteristic
697C22C2×#41011yescharacteristic
698C22#289711yescharacteristic
699C22#289711yescharacteristic
700C22#290111yescharacteristic
701C22#290111yescharacteristic
702C22#294611yescharacteristic
703C22#294611yescharacteristic
704C22#294811yescharacteristic
705C22#294911yescharacteristic
706C4—21no—
707C4—21no—
708C4—21no—
709C4—22no—
710C4—21no—
711C4—24no—
712C4—44no—
713C4—44no—
714C4—42no—
715C4—42no—
716C4—42no—
717C4—84no—
718C4—82no—
719C4—81no—
720C4—81no—
721C4—81no—
722C4—81no—
723C4C4×#9914yes—
724C2—21no—
725C2—21no—
726C2—21no—
727C2—21no—
728C2—42no—
729C2—42no—
730C2—84no—
731C2#1760311yescharacteristic
732C2#1760711yescharacteristic
733C2#1761011yescharacteristic
734C2#1761111yescharacteristic
735C2C2×C4×#9911yescharacteristic
736C2#2040511yescharacteristic
737C2#2041611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20379C2semidirect
5. #13801C2semidirect
6. #8220C2semidirect
7. #8218C2semidirect
8. #8216C2semidirect
9. C4×#327C2semidirect
10. #8016C2semidirect
11. #8012C2semidirect
13. C2×#518C4semidirect
24. #2324C4semidirect
25. #2324C4semidirect
26. #2316C4semidirect
27. #2309C4semidirect
32. #2274C4semidirect
36. #1749C22semidirect
37. C4×C22⋊C8C22semidirect
39. #1519C4semidirect
40. #1507C4semidirect
47. #1434C22semidirect
48. C4×#95C22semidirect
49. #1294C22semidirect
50. #1294C22semidirect
51. #1294C22semidirect
52. #1294C22semidirect
53. #1294C22semidirect
54. #1294C22semidirect
97. #407C2×C4semidirect
101. C2×C22⋊C8C2×C4semidirect
133. #380C2×C4semidirect
149. C4×C22⋊C4C2×C4semidirect
158. #327C2×C4semidirect
159. #327C2×C4semidirect
160. #327C2×C4semidirect
161. #327C2×C4semidirect
176. C22×D4C42semidirect
214. #121C42semidirect
242. C2×C4⋊C4C42semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#342042
#203792
C2×#15192
C2×#15072
#138012
#82202
#82182
#82162
C4×#3272
#80162
#80122